Compact-open topology and application to approximation by polynomials.
Polynomials are widely known by mathematicians and they are very useful. In many situations, we find that in topology and real analysis, we have to use continuous functions, especially continuous real-valued functions on a closed interval. The continuous functions form a very large class of functions. Often there is the need to find a smaller class of functions which are dense in the space of continuous functions. One such useful example is the classical Weierstrass approximation theorem. The purpose of this thesis is to study a modem generalization of the Weierstrass approximation theorem and find an appropriate proof for it. Stone’s theorem provides an elegant or a well-designed generalization and a proof. Therefore this project deals with Stone’s theorem and its application to Weierstrass approximation theorem. The main tool used is compact-open topology. Hence, the need for the idea of topological spaces, metric spaces, the separation axioms and compactness of the spaces.
A thesis submitted to the Mathematics Department in partial fulfilment of the requirements for the award of Masters degree in Mathematics, 2005